\documentstyle[twoside,epsf,hyperref]{article}

\input ibvs2.sty

\begin{document}
\IBVSheadDOI{63}{6227}{11 December 2017}

\IBVStitle{New Light-Time Curve of Eclipsing Binary AM L\lowercase{eo} }

\IBVSauth{Gorda, S. Yu.$^1$; Matveeva, E. A.$^2$}

\IBVSinst{Kourovka Astronomical Observatory of the Ural Federal
University; e-mail: stanislav.gorda@urfu.ru} 
\IBVSinst{Ural Federal University, 51, Lenin av., Ekaterinburg, Russia, 620000}

\SIMBADobj{AM Leo}

\IBVStyp{  Binaries: eclipsing  } 
\IBVSkey{CCD photometry, times of minima}

\IBVSabs{We present 72 photoelectric and CCD  times of minima of eclipsing binary AM Leo obtained mainly during $1996-2017$ at Kourovka Astronomical Observatory of the Ural Federal University in Russia. We obtained new values of period of 50.5 years and eccentricity of 0.28 of the orbit of the eclipsing pair around the mass center of the system AM~Leo with the third body. These results have been received taking into account the times of minima taken from literature and obtained from $JD_\odot=2435570$ to $JD_\odot=2457835$.}

\begintext

The eclipsing variable star AM~Leo (BD $+10^\circ 2234$ A) is a
bright component ($V~=~9.1 - 9.7$ mag) of the visual binary
system ADS\,8024
($\rho=11''.4$,~$\theta=270^\circ$)(Hiller et~al.\ 2004). The most
comprehensive survey of the photometric observations of AM~Leo were 
given in the studies of Hiller et~al.\ (2004) and
Albayrak et~al.\ (2005a). Many authors noted temporal variations in the light curve of
AM~Leo. Along with the light curve variations, orbital 
variations have been observed too. Various
hypotheses were proposed to explain this phenomenon. The most
likely reason for the period change in AM Leo is now considered to be
the presence of a third body in the system. This hypothesis was
first suggested by Demircan \& Derman (1992). Later
Albayrak et~al.\ (2005a) and Qian et~al.\ (2005) have determined the
parameters of the light-time curve based on the analysis of the
moments of minima from data obtained using photomultiplier and CCD
detectors only.

Albayrak et~al.\ (2005a) obtained the mutual orbital period of
AM~Leo and the third body by the very eccentric orbit to be about
45 years, they also estimated the mass of the third body to be
$M_3 = 0.18 M_\odot$. These results have been obtained on the
basis of the data collected from $JD_\odot=2435570$ to
$JD_\odot=2453106$. In the paper of Qian et~al.\ (2005) the values
of the period 51.8 years and $M_3 = 0.20 M_\odot$ were listed. But
these values were obtained with less data compared to
the paper of Albayrak et~al.\ (2005a).

Since this research a number of new values of the
moments of minima of AM~Leo have been received. In the paper of
Albayrak et~al.\ (2005a) the moments of minima were used which have
been distributed on an interval of time, corresponding only to
$\sim 1.1$ of the 45-year period. Now this interval comprises $\sim 1.4$
times of the period, and the moments of minima are distributed regularly
enough throughout. Differences O--C calculated with the new
moments of minima, already do not correspond to the light-time
curve received by Albayrak et~al.\ (2005a). Thus, now it is the
time to define again the parameters of a light-time curve of
AM~Leo.

We have obtained 72 photoelectric and CCD moments of minima of the
eclipsing binary AM~Leo generally between 1996--2017 at
Kourovka Astronomical Observatory of the Ural Federal University
in Russia, which have not been published earlier. Data were obtained 
by one of the authors with a reflector telescope ($D =
0.45$ m), equipped with a photoelectric photometer,  placed in the
Cassegrainian focus ($F=11.0$ m), and by a CCD-camera, placed in
the Newtonian focus ($F=2.0$ m).

{\normalsize
\begin{table}[ht]
\begin{center}
 \caption{Moments of minima of AM~Leo.}
\label{tpl}
\bigskip
\begin{tabular}{ccccc|ccccc}
\hline &&&&&&&&&\\[-12pt]
 Time of min. & Error& Type & Filter & Rem. &Time of min. & Error & Type & Filter & Rem. \\
 HJD 2400000+ &&&&& HJD 2400000+&&&&\\
 \hline
&&&&&&&&&\\[-10pt]
 50106.54068  & 0.00061 & II  & BV  & PE & 55594.43435  & 0.00031 & I   & BVR & CCD1 \\
 50142.39120  & 0.00205 & II  & BV  & PE & 55617.48019  & 0.00005 & I   & BVR & CCD1 \\
 50156.29112  & 0.00041 & II  & BV  & PE & 55623.33233  & 0.00084 & I   & BVR & CCD1 \\
 50156.47572  & 0.00061 & I   & BV  & PE & 55625.34501  & 0.00033 & II  & BVR & CCD1 \\
 50157.38869  & 0.00054 & II  & BV  & PE & 55630.46622  & 0.00040 & II  & BVR & CCD1\\
 50159.21672  & 0.00085 & II  & BV  & PE & 55659.36320  & 0.00114 & II  & BVR & CCD1 \\
 50159.40102  & 0.00050 & I   & BV  & PE & 55679.30038  & 0.00026 & I   & BVR & CCD1 \\
 50168.36193  & 0.00025 & II  & BV  & PE & 55953.46636  & 0.00011 & II  & BVR & CCD1 \\
 50169.27776  & 0.00015 & I   & BV  & PE & 55958.40439  & 0.00029 & I   & BVR & CCD1 \\
 53066.40291  & 0.00052 & I   & BV  & PE & 55960.41672  & 0.00017 & II  & BVR & CCD1 \\
 53090.36297  & 0.00013 & II  & BV  & PE & 55973.40238  & 0.00014 & I   & BVR & CCD1 \\
 53123.28460  & 0.00075 & II  & BV  & PE & 55978.34052  & 0.00041 & II  & BVR & CCD1 \\
 54172.39374  & 0.00010 & II  & BVR & CCD1 & 56016.20116  & 0.00025 & I   & BVR & CCD1 \\
 54208.24196  & 0.00011 & II  & BVR & CCD1 & 56016.38267  & 0.00038 & II  & BVR & CCD1 \\
 54214.27804  & 0.00021 & I   & BVR & CCD1 & 56309.38667  & 0.00037 & II  & BVR & CCD1 \\
 54459.54551  & 0.00016 & II  & BVR & CCD1 & 56309.57155  & 0.00010 & I   & BVR & CCD1\\
 54475.45769  & 0.00021 & I   & BVR & CCD1 & 56365.35475  & 0.00035 & II  & BVR & CCD1 \\
 54497.40597  & 0.00012 & I   & BVR & CCD1 & 56366.26882  & 0.00012 & I   & BVR & CCD1\\
 54537.46047  & 0.00005 & II  & BVR & CCD1 & 56385.29096  & 0.00026 & I   & BVR & CCD1 \\
 54552.27559  & 0.00043 & I   & BVR & CCD1 & 56386.38863  & 0.00093 & I   & BVR & CCD1 \\
 54571.29691  & 0.00025 & I   & BVR & CCD1 & 56400.28810  & 0.00008 & I   & BVR & CCD1\\
 54578.24718  & 0.00033 & I   & BVR & CCD1 & 56412.36013  & 0.00015 & I   & BVR & CCD1 \\
 54586.29474  & 0.00027 & I   & BVR & CCD1 & 56710.30083  & 0.00005 & II  & BVR & CCD1 \\
 54825.52650  & 0.00015 & I   & BVR & CCD1 & 56710.48448  & 0.00030 & I   & BVR & CCD1 \\
 54882.40781  & 0.00027 & II  & BVR & CCD1 & 56770.29242  & 0.00032 & II  & BVR & CCD1 \\
 54887.52890  & 0.00031 & II  & BVR & CCD1 & 56742.30875  & 0.00016 & I   & BVR & CCD1 \\
 54888.44357  & 0.00011 & I   & BVR & CCD1 & 56751.27073  & 0.00046 & II  & BVR & CCD1 \\
 54909.29397  & 0.00041 & I   & BVR & CCD1 & 57458.35613  & 0.00020 & II  & BVR & CCD2 \\
 54923.19396  & 0.00026 & I   & BVR & CCD1 & 57459.27007  & 0.00009 & I   & BVR & CCD2 \\
 55217.47827  & 0.00013 & II  & BVR & CCD1 & 57463.29403  & 0.00008 & I   & BVR & CCD2 \\
 55218.57605  & 0.00023 & II  & BVR & CCD1 & 57463.47687  & 0.00018 & II  & BVR & CCD2 \\
 55223.51509  & 0.00016 & I   & BVR & CCD1 & 57822.32412  & 0.00014 & II  & BVR & CCD2 \\
 55246.37686  & 0.00021 & II  & BVR & CCD1 & 57827.26249  & 0.00008 & I   & BVR & CCD2 \\
 55281.31090  & 0.00013 & II  & BVR & CCD1 & 57828.36025  & 0.00015 & I   & BVR & CCD2 \\
 55288.26110  & 0.00034 & II  & BVR & CCD1 & 57829.27344  & 0.00010 & II  & BVR & CCD2 \\
 55570.47523  & 0.00105 & II  & BVR & CCD1 & 57835.31003  & 0.00015 & I   & BVR & CCD2 \\
 \hline
 \end{tabular}
 \end{center}
\end{table}
}


The CCD observations data were reduced using the $MaxIm DL$ and {\it
Muniwin} (http://c-munipack.sourceforge.net) packages. The minima
time were computed by a parabola fitting method and averaged from
all filters used during the night. Values of the moments of minima
of AM~Leo, obtained from our observations, are listed in Table~1.
Abbreviation in the column named ``Rem." corresponds to the
detector used for observations:
\begin{itemize}
\item PE~---~scanning photoelectric photometer (it is not used
now); \item CCD1~---~CCD camera Apogee Alta-U6 (Kodak KAF-1001E,
1048$\times$1048, 24-micron chip); \item CCD2~---~CCD camera FLI PL230
(e2v CCD230-42-1-143, 2048$\times$2048, 15-micron chip).
\end{itemize}
Additional seven moments of minima obtained by one of the
authors in 2015 have been published by Gorda~(2016).

\IBVSfig{12cm}{6227-f1.eps}{The light-time curve of the variable
star AM~Leo (solid line); open circles denote values of the O--C
calculated from the times of minima from Albayrak et~al.\ (2005a);
open triangles represent ones from IBVS (see page 3); open squares
denote O--C  calculated from our data (see Table~1). }
\IBVSfigKey{6227-f1.eps}{AM~Leo}{O-C diagram}


For calculating the O--C differences and the parameters of the
light-time curve we used our data (see Table 1), data from the
paper of Albayrak et~al.\ (2005a), and also the moments of minima
published in {\it IBVS} from 2002 to 2017 (Pribulla et~al.\ 2002,
G\"urol et~al.\ 2003, Dvorak 2004, H\"ubscher 2005,
Albayrak et~al.\ 2005b, H\"ubscher et~al.\ 2005, Kotkov\'a \& Wolf
2006, \c{S}enavci et~al.\ 2007, Kili\c{c}o\u{g}lu~et~al.
2007,H\"ubscher 2007, Og{\l}oza et~al.\ 2007,
H\"ubscher et~al.\ 2008, Nelson 2009, Diethelm 2009,
Parimucha 2009, H\"ubscher et~al.\ 2010, Diethelm 2010,
H\"ubscher~\&~Monninger 2011, Diethelm 2011,
H\"ubscher et~al.\ 2012, Diethelm 2012, Parimucha 2013,
H\"ubscher et~al.\ 2013, Nelson 2013, H\"ubscher 2013,
Zasche 2014, H\"ubscher~\&~Lehmann 2015, H\"ubscher 2016a,
H\"ubscher 2016b, Zasche et~al.\ 2017).

{\normalsize
\begin{table}[ht]
\begin{center}
 \caption{Parameters of the light-time curve. }
\label{tpl}
\bigskip
\begin{tabular}{l|cc|cc|cc}
\hline &\multicolumn{2}{|c|}{}&\multicolumn{2}{c|}{}&\multicolumn{2}{c}{}\\[-12pt]
 &\multicolumn{2}{|c|}{Qian et al., 2005}&\multicolumn{2}{c|}{Albayrak et al., 2005a}&\multicolumn{2}{c}{This paper}\\
\hline &&&&&&\\[-12pt]
Parameter & Value & Error & Value & Error & Value & Error \\
\hline &&&&&&\\[-12pt]
N &74& &103& & 243 & \\
$\sum(O-C_{LTC})^2$& 0.00016 & & 0.00020 &  & 0.00045 &   \\
$JD_{\odot\,min\,I}$ &2439936.8260 & & 2452397.35411&  0.00006  & 2452397.35801&  0.00009  \\
$P_{orb}$ (day) &0.36579770 & & 0.365797425& 0.000000007 & 0.365797590& 0.000000008 \\
$a\sin i$ (AU) & 1.69& 0.10 & 1.36 & 0.10 & 1.30 & 0.05  \\
$e$  & 0.58  & 0.07 & 0.73& 0.04  & 0.28  & 0.03  \\
$\omega\; (^\circ)$  & 54.0& 16.6& 22.0 & 3.0 & 20.6 & 2.8  \\
$T_0$ (HJD) &2436021&859& 2436346 & 70 & 2435320 & 50  \\
$P_{12}$ (year) &51.4&  & 44.82 & 0.34  & 50.5 & 0.5  \\
$A$ (day) &0.0097& 0.0006& 0.0058  & 0.0003 & 0.0072  &  0.0008 \\
$f(m_3)$ ($M_\odot$) &0.00182&0.00033& 0.00125 & 0.00028 & 0.00086 & 0.00023\\[5pt]
 \hline
 \end{tabular}
 \end{center}
\end{table}
}

Values of parameters of the light-time curve were obtained by a
fitting method described by Gorda et~al.\ (2007). Our
fit is plotted in Fig.~1 along with the observed values. The
parameters of light-time curves obtained by Qian et~al.\ (2005),
Albayrak et~al.\ (2005a) and obtained by us are given in Table 2.
Designations in the first column of Table 2 correspond to
following parameters: $N$ is the total number of the moments of
minima under consideration, $\sum(O-C_{LTC})^2$ is the value of
the minimum sum of the squares of the residuals of O--C
differences from the light-time curve, $JD_\odot I_{min}$ and
$P_{orb}$ are reference epochs for the primary minimum and the true
period of the AM~Leo respectively, $a\sin i$, $e$, $w$, $T_0$ and
$P_{12}$ are the semi-major axis, inclination, eccentricity,
longitude and epoch of the periastron passage and the period
of the orbit of the eclipsing pair around the mass center of the
 AM~Leo system with the third body, respectively. $A$ is the
semi-amplitude of the light-time curve and $f(m_3)$ is the mass
function of the third body.

As it can be seen, the new values of only three parameters of the
AM~Leo orbit with the third body, namely $e$, $T_0$ and $P_{12}$
differ considerably from the ones received by
Albayrak et~al.\ (2005a). Our values can be considered as more
reliable at the present time because they were obtained by the
use of more data, compared to the paper of
Albayrak et~al.\ (2005a) and because our moments of minima are
distributed on the time interval exceeding the value of $P_{12}$
nearly one and a half times.

The obtained values of $P_{12} = 50.5 \pm 0.5$ and $a\sin i =
1.30\pm 0.05$ lead to a very small mass function of
$f(m_3)=0.00086 \pm 0.00023 M_\odot$ for the third body. The mass
of the third body was computed for different values of the orbital
inclination of the third body orbit and the derived values are
given in Table~3. In this computation, the masses of the
components of the eclipsing pair $M_1=1.23 M_\odot$, $M_2=0.54
M_\odot$ (Gorda 2016) were applied.

\begin{table}[ht]
\begin{center}
 \caption{Mass and semi-major axis of the third body orbit depending
on the orbital inclination. } \label{tp2}
\bigskip
\begin{tabular}{ccc}
\hline $i(^\circ)$& $m_3(M_\odot)$ & $a_3$(AU) \\
\hline
10.0 & $1.12\pm 0.13$ & $12.0\pm 1.1$\\
20.0 & $0.48\pm 0.04$ & $14.2\pm 1.2$\\
30.0 & $0.31\pm 0.03$ & $15.0\pm 1.3$\\
40.0 & $0.24\pm 0.03$ & $15.3\pm 1.4$\\
50.0 & $0.20\pm 0.02$ & $15.5\pm 1.5$\\
60.0 & $0.17\pm 0.02$ & $15.6\pm 1.5$\\
70.0 & $0.16\pm 0.01$ & $15.7\pm 1.5$\\
80.0 & $0.15\pm 0.01$ & $15.8\pm 1.6$\\
90.0 & $0.15\pm 0.01$ & $15.8\pm 1.5$\\
\hline
\end{tabular}
\end{center}
\end{table}

Below we list the light elements that can be used to compute the
period of AM~Leo for the nearest epoch of observation. We have
determined them by analyzing the moments of minima for the last 5
years. These data can be approximated quite accurately by the
following parabolic dependence:

\begin{center}
$
\begin{array}{ccrcrlccll}
JD_{\odot\,min\,I}  &=&  2452397.34402&+&0.36580143&\!\!\!\!\cdot E &-& 1.76&\!\!\!\!\cdot 10^{-10}&\!\!\!\!\cdot E^2.\\
&&\pm30&&\pm44&&&\pm32&&\\
\end{array}
$
\end{center}

We derive from that the following light elements suitable for
computing the times of minima of AM~Leo at present time:

\begin{center}
$
\begin{array}{ccrcrl}
JD_{\odot\,min\,I} &=&  24577835.30926&+&0.36579882&\!\!\!\!\cdot E.\\
%&&\pm30&&\pm44&\\
\end{array}
$
\end{center}

\noindent \textit{Acknowledgements:} This work was supported in part by the Ministry of Education and
Science (the basic part of the State assignment, RK no.
AAAA-A17-117030310283-7) and by the Act no. 211 of the Government
of the Russian Federation, agreement � 02.A03.21.0006.

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\end{document}
